Problem with applying image on a displacement map, displacement apply on top and bottom but not on sides
Try generated coordinates with Sphere projection. You do not need to UV unwrap for this.

Updated (after review file):

The question is related to applying an image texture to the Displace modifier in Blender Render. In Blender Render you can do a few more things with 'normal' textures like Sphere projection that are not available with the Displace modifier image texture - as far as I know. The distortion has to do with the way the square textures are tiled around the sphere.

The issue is further exaggerated by the scale not having being applied.



Left is the original sphere, and right with CTRL A and having scale applied. The texture was set to extend, but has been set here to repeat to avoid distortion after the scale was applied.

Some options would be to consider UV unwrapping, as Duarte suggested. This would cause seams that might be hidden by a procedural texture over this like Clouds.

The Displace modifier has been somewhat eclipsed by True Displacement which is available in Cycles Render - not Blender Render. It has the advantage that it can be used in combination with Adaptive Subdivision which also makes it less intense for your computer.

The Cycles material node system also has some versatile tools tools for achieving much of what I think you want to do. Some prefer old school Blender and I'm not judging.

I've attached the simplified file I tested - pictured above - in case someone has more solutions to offer...